Question

Taylor's lemonade stand estimates its profit P in dollars as P(x) = -4x^2 + 162x - 3, where x is the number of cups of lemonade sold. How many cups must be sold to produce the maximum profit?

A. 13 cups
B. 2 cups
C. 4 cups
D. 9 cups

Answer 1

To find the number of cups of lemonade sold for maximum profit, we calculate the vertex of the parabola given by the profit function P(x) = -4x^2 + 162x - 3, which occurs at x = 20.25, rounded to 20 cups. However, the options provided in the question do not match this calculation.

Step-by-step explanation:

The student's question involves finding the number of cups of lemonade that must be sold to produce the maximum profit, given the profit function P(x) = -4x^2 + 162x - 3. This is a quadratic function, and the number of cups that yields the maximum profit is found by determining the vertex of the parabola. The vertex of a parabola in the form of f(x) = ax^2 + bx + c occurs at x = -b/(2a). In this case, the coefficients are a = -4 and b = 162.

Applying the formula to find the vertex gives us x = -162 / (2 * -4), which simplifies to x = 20.25. Since we are dealing with whole units (cups of lemonade), we round down to the nearest whole number, which is 20 cups of lemonade, as fraction of a cup is not realistic in this context.

However, please note that this question provided four options (A. 13 cups, B. 2 cups, C. 4 cups, D. 9 cups), and none of these options is correct based on our calculation. Hence, there seems to be an error in the options provided.